3.2820 \(\int \sqrt{\frac{c}{(a+b x)^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{(a+b x) \sqrt{\frac{c}{(a+b x)^2}} \log (a+b x)}{b} \]

[Out]

(Sqrt[c/(a + b*x)^2]*(a + b*x)*Log[a + b*x])/b

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Rubi [A]  time = 0.0265199, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(a+b x) \sqrt{\frac{c}{(a+b x)^2}} \log (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c/(a + b*x)^2],x]

[Out]

(Sqrt[c/(a + b*x)^2]*(a + b*x)*Log[a + b*x])/b

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Rubi in Sympy [A]  time = 2.01883, size = 24, normalized size = 0.86 \[ \frac{\sqrt{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right ) \log{\left (a + b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c/(b*x+a)**2)**(1/2),x)

[Out]

sqrt(c/(a + b*x)**2)*(a + b*x)*log(a + b*x)/b

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Mathematica [A]  time = 0.0123437, size = 28, normalized size = 1. \[ \frac{(a+b x) \sqrt{\frac{c}{(a+b x)^2}} \log (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c/(a + b*x)^2],x]

[Out]

(Sqrt[c/(a + b*x)^2]*(a + b*x)*Log[a + b*x])/b

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Maple [A]  time = 0.007, size = 27, normalized size = 1. \[{\frac{ \left ( bx+a \right ) \ln \left ( bx+a \right ) }{b}\sqrt{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c/(b*x+a)^2)^(1/2),x)

[Out]

(b*x+a)*ln(b*x+a)*(c/(b*x+a)^2)^(1/2)/b

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Maxima [A]  time = 1.42467, size = 18, normalized size = 0.64 \[ \frac{\sqrt{c} \log \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c/(b*x + a)^2),x, algorithm="maxima")

[Out]

sqrt(c)*log(b*x + a)/b

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Fricas [A]  time = 0.216869, size = 50, normalized size = 1.79 \[ \frac{{\left (b x + a\right )} \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} \log \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c/(b*x + a)^2),x, algorithm="fricas")

[Out]

(b*x + a)*sqrt(c/(b^2*x^2 + 2*a*b*x + a^2))*log(b*x + a)/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{c}{\left (a + b x\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x+a)**2)**(1/2),x)

[Out]

Integral(sqrt(c/(a + b*x)**2), x)

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GIAC/XCAS [A]  time = 0.220453, size = 27, normalized size = 0.96 \[ \frac{\sqrt{c}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c/(b*x + a)^2),x, algorithm="giac")

[Out]

sqrt(c)*ln(abs(b*x + a))*sign(b*x + a)/b